How Absolute Value and Step Functions Use Piecewise Rules

Absolute value is already a piecewise rule

Absolute value measures distance from zero, so its output cannot be negative. Algebraically,

$$|u|= \begin{cases} u, & u\ge0,\\ -u, & u<0. \end{cases}$$

For $f(x)=|x-h|$, the expression inside the bars changes sign at $x=h$. That input is the vertex and the boundary between two linear pieces. The graph falls toward $(h,0)$ from the left and rises away from it on the right.

Restricting the domain can also change the range. To find a range on $a\le x\le b$, check the vertex when it lies inside the interval and compare the endpoint outputs. The smallest and largest of those candidates determine the output interval.

Equivalent expressions must agree for every input

Visual similarity is not enough to prove two absolute-value expressions are equivalent. The bars change the order of operations: $|x-c|$ takes a distance after subtracting, while $|x|-c$ subtracts after taking a distance.

To disprove equivalence, find one counterexample. Inputs near zero or near the point where the inside expression changes sign are often efficient. If the two outputs differ once, the expressions are not identical functions.

Step functions assign one output to each interval

A step function is a piecewise function whose formula is constant on each interval. A graph shows each piece as a horizontal segment. The endpoint symbols carry mathematical information:

• a closed point means the boundary input is included;
• an open point means it is excluded;
• at a shared boundary, exactly one piece should usually own the input so the relation remains a function.

That ownership must match the inequalities. For example, $a\le x<b$ is closed at $a$ and open at $b$.

The range of a step function is usually a set, not an interval

If a pricing rule can output only 0.50, 0.70, 0.90, or 1.15, then its range is

$$\{0.50,0.70,0.90,1.15\}.$$

Writing $0.50\le y\le1.15$ would incorrectly include values such as 0.80 that the function never produces. Domain intervals can be continuous while the range remains discrete.

A reliable graphing routine

For each piece, first mark its interval on the $x$-axis, draw the horizontal or linear rule only over that interval, and then add open or closed endpoints. Finally, test every shared boundary directly in the written definition. This last check catches the most common mistake: assigning two outputs—or no output—to the same boundary input.